Lens based far-field imaging is limited in the resolution that it can achieve by the characteristics of the lens. In general, there are problems of diffraction of the lens, problems with aberration of the lens and problems of out-of-focus radiation. The latter, out-of-focus radiation problem is generally partially improved by the use of confocal imaging methodologies; in optics, non-linear imaging techniques are also useful. The solution of the former diffraction and aberration problems is partially addressed by measuring the point spread function of the lens and then using computer deconvolution to remove these effects from the image. Even the latter out-of-focus problem can be addressed without confocal or non-linear imaging by considering both the in-focus and the out-of-focus point spread function and using deconvolution routines to try and eliminate these effects. Numerous algorithms have been devised to address these problems of computer deconvolution of far-field imaging data, but none are completely successful and none of them have the ability to carry the far-field image to the realm beyond the diffraction limit as defined, for example, by the Rayleigh criterion, which is approximately ½ of the wavelength of the radiation that is being used. For visible 500 nm light this is 250 nm.
In terms of deconvolution algorithms, a powerful mathematical approach is based on the use of constraints. For example, in deconvolving a far-field image a good constraint would be to define with high precision the cell membrane of a cell that is stained with a dye and is being imaged by a lens. By precisely defining the position of a cell membrane or a portion of the cell membrane, it is possible precisely to define where the staining in the image is confined and beyond which point or points there is no staining and its associated optical phenomenon. Such a constraint would give many deconvolution algorithms a powerful advantage. Nonetheless, even though the idea is mathematically a powerful concept [Carrington et al. Science 268, 1483 (1995)], it is seriously limited in far-field optics by the inability to obtain a constraint that is better than the optical resolution.